Integration by Parts Math Example 3

Solution

1. 1
   Step 1: First IBP — let $u = x^2$, $dv = e^x\,dx$; then $du = 2x\,dx$, $v = e^x$.
2. 2
   Step 2: Apply the formula: $\int x^2 e^x\,dx = x^2 e^x - \int 2x e^x\,dx$.
3. 3
   Step 3: Second IBP on $\int 2x e^x\,dx$ — let $u = 2x$, $dv = e^x\,dx$; then $du = 2\,dx$, $v = e^x$.
4. 4
   Step 4: $\int 2x e^x\,dx = 2x e^x - \int 2 e^x\,dx = 2x e^x - 2e^x$.
5. 5
   Step 5: Combine: $\int x^2 e^x\,dx = x^2 e^x - (2x e^x - 2e^x) = x^2 e^x - 2x e^x + 2e^x + C = e^x(x^2 - 2x + 2) + C$.